Coherence via Focusing for Symmetric Skew Monoidal Categories

Veltri, Niccolò

doi:10.1007/978-3-030-88853-4_12

Cited by 3 publications

(8 citation statements)

References 26 publications

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“…A sequent is a triple of the form S | Γ ⊢ A, where the succedent A is a single formula (as in NMILL) and the antecedent is divided in two parts: an optional formula S, called stoup [10], and an ordered list of formulae Γ, called context. The peculiar design of sequents, involving the presence of the stoup in the antecedent, comes from previous work on deductive systems with skew structure by Uustalu, Veltri and Zeilberger [26,25,24,27]. The metavariable S always denotes a stoup, i.e., S can be a single formula or empty, in which case we write S = −, and X ,Y, Z are always names of atomic formulae.…”

Section: A Sequent Calculus For Skew Non-commutative Millmentioning

confidence: 99%

“…Alternatively, following the strategy of our previous papers [26,25,24,27], this result can be shown by going via a Hilbert-style deductive system that directly presents the free skew monoidal closed category on At. Its formulae are the same of the sequent calculus, its sequents are pairs A ⇒ B, with A and B single formulae.…”

Section: Categorical Semantics Via Skew Monoidal Closed Categoriesmentioning

confidence: 99%

“…This paper represents the latest installment of a large project aiming at the development of the proof theory of categories with skew structure. So far the project, promoted and advanced by Uustalu, Veltri and Zeilberger, has investigated proof systems for skew semigroup categories [28], (non-symmetric and symmetric) skew monoidal categories [26,25,27] and skew prounital closed categories [24]. From a purely proof-theoretic perspective, the main lesson gained from the study of these skew systems consists in first insights into ways of modular construction of focusing calculi.…”

Section: A Focused System With Tag Annotationsmentioning

confidence: 99%

“…The design of the sequent calculus draws inspiration from, and further advance, the line of work of Uustalu, Veltri and Zeilberger on proof systems for various categories with skew structure: skew semigroup (a.k.a. the Tamari order) [28], skew monoidal (non-closed) [26,25] and its symmetric variant [27], skew prounital (non-monoidal) closed [24].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Proof Theory of Skew Non-Commutative MILL

Uustalu¹,

Veltri²,

Wan³

2022

Electron. Proc. Theor. Comput. Sci.

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Monoidal closed categories naturally model NMILL, non-commutative multiplicative intuitionistic linear logic: the monoidal unit and tensor interpret the multiplicative verum and conjunction; the internal hom interprets linear implication. In recent years, the weaker notion of (left) skew monoidal closed category has been proposed by Ross Street, where the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation. A question arises: is it possible to find a logic which is naturally modelled by skew monoidal closed categories? We answer positively by introducing a cut-free sequent calculus for a skew version of NMILL that is a presentation of the free skew monoidal closed category. We study the proof-theoretic semantics of the sequent calculus by identifying a calculus of derivations in normal form, obtained from an adaptation of Andreoli's focusing technique to the skew setting. The resulting focused sequent calculus peculiarly employs a system of tags for keeping track of new formulae appearing in the antecedent and appropriately reducing non-deterministic choices in proof search. Focusing solves the coherence problem for skew monoidal closed categories by exhibiting an effective procedure for deciding equality of maps in the free such category.

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Section: A Sequent Calculus For Skew Non-commutative Millmentioning

confidence: 99%

Section: Categorical Semantics Via Skew Monoidal Closed Categoriesmentioning

confidence: 99%

Section: A Focused System With Tag Annotationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Proof Theory of Skew Non-Commutative MILL

Uustalu¹,

Veltri²,

Wan³

2022

Electron. Proc. Theor. Comput. Sci.

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“…In our previous investigations we have explored deductive systems for (i) skew semigroup [26], (ii) skew monoidal [23], (iii) skew (prounital) closed [21] and (iv) skew monoidal closed categories [20,25], corresponding to skew variants of the fragments of non-commutative intuitionistic linear logic consisting of connectives (i) ⊗, (ii) (I, ⊗), (iii) ⊸ and (iv) (I, ⊗, ⊸). We have also studied partial normality conditions, when one or more among associator and unitors is allowed to have an inverse [22], and extensions with exchange à la Bourke and Lack [24].…”

Section: Introductionmentioning

confidence: 99%

Semi-Substructural Logics with Additives

Veltri,

Wan

2024

Electron. Proc. Theor. Comput. Sci.

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This work concerns the proof theory of (left) skew monoidal categories and their variants (e.g. closed monoidal, symmetric monoidal), continuing the line of work initiated in recent years by Uustalu et al. Skew monoidal categories are a weak version of Mac Lane's monoidal categories, where the structural laws are not required to be invertible, they are merely natural transformations with a specific orientation. Sequent calculi which can be modelled in such categories can be identified as deductive systems for restricted substructural fragments of intuitionistic linear logic. These calculi enjoy cut elimination and admit a focusing strategy, sharing resemblance with Andreoli's normalization technique for linear logic. The focusing procedure is useful for solving the coherence problem of the considered categories with skew structure.Here we investigate possible extensions of the sequent calculi of Uustalu et al. with additive connectives. As a first step, we extend the sequent calculus with additive conjunction and disjunction, corresponding to studying the proof theory of skew monoidal categories with binary products and coproducts satisfying a left-distributivity condition. We introduce a new focused sequent calculus of derivations in normal form, which employs tag annotations to reduce non-deterministic choices in bottom-up proof search. The focused sequent calculus and the proof of its correctness have been formalized in the Agda proof assistant. We also discuss extensions of the logic with additive units, a form of skew exchange and linear implication.

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Maximally Multi-focused Proofs for Skew Non-Commutative MILL

Veltri

2023

Lecture Notes in Computer Science

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Coherence via Focusing for Symmetric Skew Monoidal Categories

Cited by 3 publications

References 26 publications

Proof Theory of Skew Non-Commutative MILL

Proof Theory of Skew Non-Commutative MILL

Semi-Substructural Logics with Additives

Maximally Multi-focused Proofs for Skew Non-Commutative MILL

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